Integration Worksheet5 solution8

In this page integration worksheet5 solution8 we are going to see solution of some practice question from the worksheet of integration.

Question 27

Integrate the following with respect to x,x (x - a)^m

Solution:

                = ∫ x (x - a)^m dx

let t = x - a

differentiating with respect to "x"

dt = dx - 0

dt = dx

x = t + a

                = ∫ x (x - a)^m dx

                = ∫ (t + a) t^(m) (dt)

                = ∫ [t^(m + 1) - a t^(m)] dt

                = ∫ [t^(m + 1) - at^(m)] dt

                = ∫ t^(m + 1) dt - ∫ at^(m) dt

                = t^(m + 2)/(m + 2) - a t^(m + 1)/(m + 1) + C

now we are going to apply the value of t

                 = (x - a)^(m + 2)/(m + 2) - a t^(m+1)/(m+1) + C


Question 28

Integrate the following with respect to x, x² (2 - x)^15

Solution:

                = ∫ x (x - a)^m dx

let t = x - a

differentiating with respect to "x"

dt = dx - 0

dt = dx

x = t + a

                = ∫ x (x - a)^m dx

                = ∫ (t + a) t^(m) (dt)

                = ∫ [t^(m + 1) - a t^(m)] dt

                = ∫ [t^(m + 1) - at^(m)] dt

                = ∫ t^(m + 1) dt - ∫ at^(m) dt

                = t^(m + 2)/(m + 2) - a t^(m + 1)/(m + 1) + C

now we are going to apply the value of t

                 = (x - a)^(m + 2)/(m + 2) - a t^(m+1)/(m+1) + C


Question 29

Integrate the following with respect to x, sin√x/√x

Solution:

                = ∫ sin√x/√x dx

let t = √x

differentiating with respect to "x"

    dt = (1/2√x) dx

  2 dt = (1/√x) dx

                = ∫ sin√x/√x dx

                = ∫ sin t (2 dt)

                =  2∫ sin t dt

                =  2 (- cos t) + C

                =  - 2 cos t + C

now we are going to apply the value of t

                 = -2 cos √x + C

integration worksheet5 solution8 integration worksheet5 solution8